2019年度秋季総合分科会(於:金沢大学)

SUMMARY: The present talk deals with Ulam stability for the diamond-alpha difference equation \[ \Diamond _\alpha x(t) - \lambda x(t) = 0, \quad \alpha \in [0,1], \; t\in \mathbb {Z}, \] where \(\lambda \in \mathbb {R}\) and \[ \Diamond _\alpha x(t):=\alpha \Delta x(t) + (1-\alpha )\nabla x(t). \] Note here that \(\Delta x(t)\) and \(\nabla x(t)\) mean forward difference \(x(t+1)-x(t)\) and backward difference \(x(t)-x(t-1)\), respectively. The purpose of this talk is to find an explicit Ulam stability constant for the diamond-alpha difference equations.

SUMMARY: We study the existence and location of the resonances of a \(2\times 2\) semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels cross at some point,and one of them is bonding (trapping), while the other one is anti-bonding (non-trapping). Considering energy levels just above that of the crossing, we find the asymptotics of both the real parts and the imaginary parts of the resonances close to such energies. This is a continuation of our previous works where we considered energy levels around that of the crossing. This talk is based on joint works with S. Fujiié (Ritsumeikan) and A. Martinez (Bologna).

SUMMARY: Bernoulli’s free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. This problem arises as the Euler–Lagrange equation for minimizing the capacity among all subsets of equal volume in a prescribed container. There exist two different types of solutions: elliptic and hyperbolic solutions. Elliptic solutions are “stable” solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are “unstable” solutions of which the qualitative behavior is less known. I will present an implicit function theorem based on the parabolic maximal regularity, which enables us to handle the so-called loss of derivatives without losing the regularity of solutions. As an application, we prove the existence of a foliated family of hyperbolic solutions.

SUMMARY: The asymptotic behavior of solutions to the damped wave equation has been studied for a long time after a pioneering work by Matsumura (1976). He proved \(L^p\)-\(L^q\) estimates for the damped wave equation (so-called Matsumura estimates) and applied them to semilinear problems. After that, Nishihara (2003) discovered a decomposition of the solution into the heat part and the wave part, which gives a refined \(L^p\)-\(L^q\) estimates. In this talk, we give a survey of the study of the asymptotic behavior of solutions to the damped wave equation, and show sharp \(L^p\)-\(L^q\) estimates with derivative loss. Moreover, as an application of \(L^p\)-\(L^q\) estimates, we consider the Cauchy problem of the nonlinear damped wave equation with slowly decaying initial data. In particular, we give a small data global existence result including the case of critical nonlinearity. This result is based on a joint work with M. Ikeda, T. Inui, and M. Okamoto. At the end of the talk, as another application, we also introduce Strichartz estimates for the damped wave equation including the endpoint case. This part is based on a joint work with T. Inui.

SUMMARY: As a continuation work, we consider a second order nonlinear differential equation denoted in the title. We show the domains of its solutions and have analytical expressions valid in the neighbourhoods of the ends of these domains. In this way, we clarify asymptotic behaviour of all solutions.

SUMMARY: In this talk, we consider the damped half-linear differential equation \((\varPhi _p(x'))'+a(t)\varPhi _p(x')+b(t)\varPhi _p(x)=0,\) where the coefficients \(a\) and \(b\) are periodic functions; the real-valued function \(\varPhi _p\) is the real-valued function defined by \(\varPhi _p(u) = |u|^{p-2}u\) for \(u \not = 0\) and \(\varPhi _p(0) = 0\).

The purpose of this talk is to give new criteria which guarantee that all non-trivial solutions of the damped half-linear differential equation are oscillatory (or nonoscillatory).

SUMMARY: In this talk we consider a linear integral system with two delays. We present some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable by using analysis of characteristic roots. We also investigate the limit of solutions in the critical case where the system loses its asymptotic stability.

SUMMARY: The objective of this talk is to deepen the understanding of the connection between the continuous and smooth dependence of solutions on initial conditions and the regularity of the history functionals for retarded functional differential equations. We consider some differential equation with a single constant delay with the history space of \(L^p\)-type and obtain the above dependence result by assuming the growth rate of the nonlinearity and its derivative. The corresponding history functional is discontinuous, and it becomes clear that there are the continuity and the smoothness of the composition operators (also called the superposition operators or the Nemytskii operators) between \(L^p\)-spaces behind the dependence results.

SUMMARY: Using the variational method, Chenciner and Montgomery proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics. We study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits.

SUMMARY: Second-order quasilinear ordinary differential equations with critical coefficients are considered. Asymptotic forms of slowly decaying solutions of such equations are determined.

SUMMARY: We study the bifurcation problems of semilinear ordinary differential equations with special oscillatory nonlinearities. Since \(\lambda = \lambda (\alpha )\) is a continuous function of \(\alpha > 0\), we are interested in the global behavior of \(\lambda (\alpha )\). Here, \(\alpha \) is the maximum norm \(\alpha = \Vert u_\lambda \Vert _\infty \) of the solution \(u_\lambda \) associated with \(\lambda \). In the main theorem, we obtain the precise asymptotic behavior of \(\lambda (\alpha )\) as \(\alpha \to \infty \).

SUMMARY: Voros coefficients are important objects in the exact WKB analysis for the global study of solutions of differential equations. In this talk I will report that the Voros coefficients for hypergeometric differential equations associated with 2-dimensional Garnier systems are given by the generating functions of free energies defined in terms of Eynard and Orantin’s topological recursion.

SUMMARY: Topological recursion was originally formulated as an algorithm to compute the large \(N\) expansion of correlation / partition function of matrix models from their spectral curves. I will apply the topological recursion to a family of genus 1 spectral curves, and show that the discrete Fourier transform (with respect to the period of the spectral curve) of the topological recursion partition function gives the \(\tau \)-function of the first Painlevé equation. The result is based on a relationship between the topological recursnion and the WKB analysis.

SUMMARY: We investigate spectral theory for one-body Stark Hamiltonian under minimum regularity and decay condition on the potential. Our results are proved in sharp form employing Besov-type spaces. For the proofs we adopt a new commutator scheme by Ito–Skibsted. A feature of this scheme is a particular choice of an escape function related to the classical mechanics. The whole setting, such as the conjugate operator and the Besov-type spaces, is generated by this single escape function. This talk is based on a joint work with T. Adachi, K. Ito and E. Skibsted.

SUMMARY: On a large scale free network, it has been observed that its graph Laplacian eigenvectors localize on the nodes with similar degrees. By using the graphon theory, the continuum limit of the graph Laplacian of scale free networks is a self adjoint operator. In the talk, we show that the operator is sectorial through determining its spectra, and the maximum principle holds. As a consequence, we verify that the singularity of eigenfunction-like objects of its continuous spectra is the origin of localization.

SUMMARY: In this paper, we establish a weighted version of Hardy’s inequality and improve it by adding sharp remainder terms. As weight functions we consider power type weights \(t^{\alpha p}\) for \(t\in [0,1]\). Surprisingly our result on this matter is essentially dependent on the range of parameter \(\alpha \).

SUMMARY: We consider a maximization problem on the Trudinger–Moser inequality with compact term. In this talk we study condition of the compact term on existence and nonexistence.

SUMMARY: As one of O’Hara’s energies, the Möbius energy was named after its invariant property under Möbius transformations of the surrounding space. Doyle and Schramm gave an expression of the Möbius energy in terms of the cosine of conformal angle, called the cosine formula. Since the conformal angle is Möbius invariant, we can see easily the invariant property of the Möbius energy from the formula. In this talk, an analogue of the cosine formula holds for generalized O’Hara’s energies in spite of lack of the Möbius invariant property. This newfound formula shows quantitatively how far the energy is from the Möbius invariance.

SUMMARY: We study the eigenvalue problem of the second order elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Therefore, analyzing the properties of the eigenvalue we can retrieve information on the frequency of the oscillations. Particularly, we deal with a thin rod with axial symmetry and clamped ends. It is known that there are many low-frequency eigenvalues corresponding to the bending mode of vibrations. We see as well that there appear mid-frequency eigenvalues corresponding to torsional and stretching modes of vibrations. We investigate the asymptotic behavior of these mid-frequency eigenvalues, we obtain a characterization formula of the limit equation when the thinness parameter tends to 0 and we give a result on the strong convergence of the corresponding eigenfunctions.

SUMMARY: In this talk, we consider the one-dimensional Schnakenberg model on the interval \((-1, 1)\) with periodic heterogeneity \(g(x)\). Let \(N \ge 1\) be an arbitrary natural number. We assume that \(g(x)\) is a symmetric and periodic function, namely \(g(x)=g(-x)\) and \(g(x)=g(x+2N^{-1})\). Furthermore, we assume that \(g(x)>0\) and \(g\in C^3(-1,1)\). We study the linear stability of \(N\)-peak symmetric stationary solutions. We reveal the effect of the periodic heterogeneity on the stability of \(N\)-peak solution. In particular, we investigate how \(N\)-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case \(g(x)=1\).

SUMMARY: In this talk, we consider the one-dimensional Schnakenberg model on the interval \((-1, 1)\) with heterogeneity \(g(x)\). We first construct one-peak stationary solutions. Next, we study the stability of this solution. Also, we give some condition related to the existence of one-peak solution. Since \(g(x)\) may be not symmetric on the interval \((-1,1)\), the constructed solution may be not symmetric. In particular, we reveal the effect of the heterogeneity on the location of a concentration point and the stability.

SUMMARY: In this lecture, we study the existence of positive radial solutions for a semipositone elliptic equation with a parameter \(\lambda >0\). We give a weak and general sufficient condition on \(f\) for the existence of positive radial solutions when \(\lambda >0\) is large and for the nonexistence of positive radial solutions when \(\lambda >0\) is small.

SUMMARY: We study the existence problem for positive solutions \(u\) to the quasilinear elliptic equation \[ -\Delta _{p} u = \sigma u^{q} + \mu \] in the sub-natural growth case \(0 < q < p - 1\), where \(\Delta _{p}u = \nabla \cdot ( |\nabla u|^{p - 2} \nabla u )\) is the \(p\)-Laplacian with \(1 < p < \infty \) and \(\sigma , \mu \) are nonnegative measurable functions (or measures) on \(\mathbb {R}^{n}\). We construct solutions in Lorentz spaces with a sharp exponent. To derive existence of such solutions, we give estimates for generalized mutual energy of \(\sigma \) and \(\mu \). Our method can be applied for equations with several subnatural terms.

SUMMARY: We study the uniqueness of positive radial solutions of \[ \Delta u(x)+f(u(x))=0\quad \mbox {in $A_{a,b}$}, \qquad u(x)=0\quad \mbox {on $\partial A_{a,b}$,} \] where \(N\geq 2\), \(A_{a,b}=\{x\in \mathbb {R}^N:a<\lvert x\rvert <b\}\). By changing a variable appropriately, we can transform the problem to the following two point boundary value problem \[ v_{ss}(s)=g(s,v(s)),\quad s\in (\alpha ,\beta ), \qquad \quad v(\alpha )=v(\beta )=0. \] We study the uniqueness of positive solution of the latter problem, and we apply it to the former problem.

SUMMARY: We show uniform boundedness of the solution to reaction diffusion equation with quadratic growth provided with mass dissipation. This property holds if the space dimension \(n\leq 3\), and for any dimension under the additional assumption of entropy inequality.

SUMMARY: In 1952, Prof. A. Turing has reported a novel principle of pattern formation, so called Turing Instability nowadays, and he has theoretically shown that spatially structured pattern is created out of obvious uniformed state spontaneously. Classically and typically, RD-equation system of Activator-Inhibitor type nonlinearity is well-known to have such an interesting property. Especially, if the diffusion constant of activator is very small, then plenty of stable steady states exit (for instance, see Y. Nishiura’s report in Dynamics reported 3 (new series)). Today, I reported that, if the time constant of inhibitor is also equal to 0, then the system has an effective energy by which the system can be regarded as a gradient system , and moreover, the most stable steady state is characterized by use of it. I will report it as a mathematically rigorously proved theorem which is based on the collabolation with Prof. Y. Nishiura (AIMR, Tohoku Univ.).

SUMMARY: We discuss the ergodic problem for viscous Hamilton–Jacobi equations with superlinear Hamiltonian, inward-pointing drift, and positive potential function vanishing at infinity. Under some radial symmetry of the drift and the potential outside a bounded region, we establish sharp estimates of the generalized principal eigenvalue with respect to a perturbation of the potential. It turns out that the asymptotic behavior of the generalized principal eigenvalue depends sensitively on the intensity of the inward drift as well as the decay order of the potential function.

SUMMARY: Let \(\{H_t\}\) be a Hamilton–Jacobi semigroup acting on functions that are bounded and uniformly continuous on \(\mathbb {R}\). Let \(\tau \) be the Takagi function. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on \(\mathbb {R}\).

Our aim of this talk is to show that the flow \(\{H_{t}\tau \}\) has a self-affine property of evolutional type inheriting a self-affine property of \(\tau \).

SUMMARY: We consider the existence of local and global-in-time strong solutions to the advection-diffusion equation with variable coefficients on an evolving surface with a boundary. We show the existence of local and global-in-time strong solutions to the advection-diffusion equation. Moreover, we derive the asymptotic stability of the global-in-time strong solution.

SUMMARY: Recently, some evolution equation related to grain boundary motion with dynamic lattice misorientations has been proposed by Epshteyn–Liu–Mizuno. The grain boundary moves by its mean curvature with time-dependent non-local mobility function. We show the existence of the classical solutions for the evolution equation when the grain boundary is described by a graph. Key tools are a priori gradient estimates, which is derived from the so-called monotonicity formula of Huisken type. We establish the monotonicity formula for the length element of the equation.

SUMMARY: We consider the \(H^{-m}\) gradient flow of length for closed plane curves. This flow is a generalization of curve diffusion flow. For the flow, evolving curves may develop singularities in finite time even if the initial curve is smooth. Furthermore, very little appears to be known regarding sufficient conditions for global existence. Hence we investigate the large-time behavior assuming the global existence of the flow. Then we show that the evolving curve converges exponentially to a circle. To do this, we use interpolation inequalities between the deviation of curvature and the isoperimetric ratio, recently established by Nagasawa and the author.

SUMMARY: In this talk, we consider the asymptotic limit of diffused surface energy in the van der Waals–Cahn–Hillard theory when an advection term is added and the energy is uniformly bounded. We show that the limit interface as \(\varepsilon \) tend to zero is an integral varifold and the generalized mean curvature vector is determined by the advection term. As an application of our result, a prescribed mean curvature problem is solved using the min-max method.

SUMMARY: We study the traveling waves for surface diffusion of plane curves. We consider an evolving plane curve with two endpoints which can move freely on the \(x\)-axis with generating constant contact angles. For the evolution of this plane curve governed by surface diffusion, we discuss the existence, the uniqueness and the convexity of traveling waves. The main results show that the uniqueness and the convexity can be lost depending on the conditions of the contact angles, although the existence holds for any contact angles in the interval \((0, \pi /2)\).

SUMMARY: For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre (2007). This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for imbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation.

SUMMARY: Radially symmetric solutions of a semilinear heat equation \(u_{t}=\Delta u + u^{p}\) on the hyperbolic space are considered. First universal bounds of the nonnegative solution are obtained to know the blow-up rate at the final blow-up time under the exponent \(p\) which is subcritical in the Sobolev sense. Next we derive its local blow-up profile and also analyze blow-up set of solutions.

SUMMARY: We consider solutions of quasilinear equations \(u_{t}=\Delta u^{m} + u^{p}\) in \(\mathbb {R}^{N}\) with the initial data \(u_{0}\) satisfying \( 0 < u_{0}< M\) and \(\lim _{|x|\to \infty }u_{0}(x)=M\) for some constant \(M>0\). It is known that, if \(0<m <p\) with \(p>1\), blow-up occurs only at space infinity. In this paper, we find solutions \(u\) that blow up throughout \(\mathbb {R}^{N}\) when \(m>p>1\).

SUMMARY: We discuss the existence of type II blowup solutions for the energy critical heat equation in 5D and 6D. Our main tool is inner-outer gluing method developed by del Pino–Musso–Wei and their collaborators.

SUMMARY: In this talk, I will present a blow-up result for \(k\)-equivariant harmonic map heat flow from \(\mathbb {R}^d\) to a unit sphere \(\mathbb {S}^d \subset \mathbb {R}^{d+1}\). We prove constructively the existence ofasymptotically non-self-similar blow-up solutions with precise description of their local space-time profiles. The blow-up solutions arise from, depending on the combination of \(d\) and \(k\), two different approximations of the nonlinear term: either through a Dirac mass supported at the origin or via a Taylor expansion around equator map \(u=\pi /2\). Transition of the blow-up mechanisms arises, accordingly.

SUMMARY: We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state.

SUMMARY: We study the reaction-diffusion system with coupled exponential nonlinearities \[ \begin{cases} \partial _t u=\Delta u+e^{p_1 u+p_2 v} & \textrm {in $\mathbb {R}^N \times (0,T)$,} \\ \partial _t v=\Delta v+e^{q_1 u+q_2 v} & \textrm {in $\mathbb {R}^N \times (0,T)$,} \\ u(x,0)=u_0 (x), v(x,0)=v_0 (x) & \textrm {in $\mathbb {R}^N$,} \end{cases} \] where \(N\ge 1\), \(T>0\), \(p_i \ge 0\) and \(q_i \ge 0\) \((i=1,2)\) with \((p_1, p_2)\neq (0,0)\) and \((q_1, q_2)\neq (0,0)\). The initial functions \(u_0\) and \(v_0\) are nonnegative and measurable. For each \((p_1, p_2, q_1, q_2)\), we obtain integrability conditions of \((u_0,v_0)\) which explicitly determine the existence/nonexistence of a local in time nonnegative classical solution. Our analysis can be applied to other nonlinearities including superexponential ones.

SUMMARY: In this talk, we obtain the precise description of the large time behavior of ODE type solutions by use of the solutions to the heat equation and reveal the relationship between the behavior of the solution and the diffusion effect nonlinear parabolic system has.

SUMMARY: We consider the Cauchy problem for a fourth order semilinear parabolic equation \(\partial _t u+(-\Delta )^2u=-\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) on \({\bf R}^N\), where \(p>2\) and \(N\ge 1\). In this talk we give a sufficient condition for the existence of solution \(u\) to the Cauchy problem such that its maximal existence time \(T_M(u)\) is finite. We prove that, if \(T_M(u)<\infty \), then the following hold:
 (a) \(\|\nabla u(t)\|_{L^\infty ({\bf R}^N)}\) blows up at \(t=T_M(u)\) for \(p>2\);
 (b) \(\|u(t)\|_{L^\infty ({\bf R}^N)}\) blows up at \(t=T_M(u)\) for \(2<p<4\).
In this talk we will show you more precise statement including the lower bound of blow up rate.

SUMMARY: This talk is concerned with the obstacle problem for a fourth order semilinear parabolic equation. Formally, the parabolic obstacle problem can be regarded as the \(L^2\)-gradient flow for an energy functional under a constraint by the obstacle. However, since the obstacle generally causes a lack of regularity of solutions, it is not clear that the obstacle problem has a gradient structure of the energy functional. In this talk, we prove that (i) the obstacle problem possesses a unique weak solution; (ii) the weak solution has the \(L^2\)-gradient structure for the energy functional in a weak sense.

SUMMARY: This talk is concerned with solvability of a quasilinear degenerate chemotaxis system with flux limitation. In a special setting Bellomo–Winkler proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions in 2017. However, a general setting has not been considered yet. The purpose of the present talk is to derive local existence and extensibility criterion ruling out gradient blow-up in a slightly general setting, and moreover to show global existence and boundedness of solutions under some conditions.

SUMMARY: This talk is concerned with blow-up of solutions to a quasilinear degenerate chemotaxis system with flux limitation. In a special setting Bellomo–Winkler found initial data such that a corresponding solution blows up in finite time in 2017. On the other hand, recently, local existence and extensibility criterion ruling out gradient blow-up in a general setting was proved; however, blow-up solutions in the general setting has not been studied yet. The purpose of the present talk is to give some conditions for existence of blow-up solutions in the general setting.

SUMMARY: We consider the Cauchy problem for an attraction-repulsion chemotaxis system in the whole space: \(\partial _tu=\Delta u-\nabla \cdot (u\nabla (\beta _1v_1-\beta _2v_2))\), \(0=\Delta v_1-\lambda _1v_1+u\), \(0=\Delta v_2-\lambda _2v_2+u\), where the constants \(\beta _1\), \(\beta _2\), \(\lambda _1\), \(\lambda _2\) are positive and the initial data \(u_0\) is nonnegative. In this talk we will discuss the global existence and blow up for this system under the condition \(\beta _1=\beta _2\).

SUMMARY: We study the drift-diffusion equation with fractional dissipation \((-\Delta )^{\theta /2}\) arising from a model of semiconductors. First we prove the existence of the small solution to the corresponding stationary problem in the whole space. Moreover it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data is suitably close to the stationary solution and the stationary solution is sufficiently small.

SUMMARY: The initial value problem of the quasi-geostrophic equation is studied. Upon the suitable conditions for the initial data, global existence in time of solutions is known. Sharp estimates for decay of solutions as the spatial parameter tends to infinity are shown.

SUMMARY: The time-global existence of unique smooth positive solutions to the reaction diffusion equations of the Keener–Tyson model for the Belousov–Zhabotinsky reaction in the whole space is established with bounded non-negative initial data. Deriving estimates of semigroups and time evolution operators, and applying the maximum principle, the unique existence and the positivity of solutions are ensured by construction of time-local solutions from certain successive approximation.

SUMMARY: The purpose in this talk is to determine the global behavior of solutions to the initial-boundary value problems for the focusing energy-subcritical and critical semilinear heat equations by initial data at low energy level in various situations by a unified treatment.

SUMMARY: Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces. This talk is based on joint-work with Neal Bez and Sanghyuk Lee.

SUMMARY: We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators on the scale of Sobolev spaces from the existence and completeness of the ordinary wave operators. Some applications to the potential scattering on the Euclidean space as well as the scattering for a nonlinear Schrödinger equation with a linear potential are also discussed. The class of potentials satisfying our condition in case of the Sobolev space of order one includes short-range potentials with subcritical singularities, the inverse-square potential and the 1D delta type point interaction.

SUMMARY: As a generalization of Carlson’s problem, Cho–Lee–Vargas considered the pointwise convergence problem for the solution of the standard Schrödinger equation along directions determined by a given compact subset of the real line. We simplify and extend their result to fractional Schrödinger equations by avoiding the use of a time localization lemma.

SUMMARY: We determine the \(H^s\) wave front sets of solutions to time dependent Schrödinger equations with a sub-quadratic potential by using the characterization of the \(H^s\) wave front set in terms of wave packet transform which is obtained by K. Kato, M. Kobayashi, and S. Ito (2017).

SUMMARY: We establish the energy estimate for higher order linear Schrödinger type equations on the torus. The proof is based on the energy method with correction terms, but some derivative losses cannot be recovered and they may have an affect on the well-posedness. As a corollary, we can classify the Cauchy problem into three types: dispersive type, parabolic type and ill-posed type.

SUMMARY: We deal with the quadratic nonlinear Schrödinger system in five dimensions. We consider the scattering solutions with the initial data below the ground state. When the system has the mass-resonance condition, first speaker has already given the sufficient and necessary condition. In this talk, we consider the system without the mass-resonance condition. We give a sufficient condition. We remark that if the system does not have the mass-resonance condition, then there is no Galilean transform invariance. We assume that the solutions are radially symmetric instead.

SUMMARY: We consider the asymptotic behavior of solutions to the nonlocal nonlinear Schrödinger equation with dissipative nonlinearity. We prove that there exists a solution which has different behavior from that of the typical cubic nonlinear Schrödinger equation.

SUMMARY: We construct a finite time blow up solution for the nonlinear Schrödinger equation with the power nonlinearity whose coefficient is complex number. We generalize the range of both the complex coefficient and the power for the result of Cazenave, Martel and Zhao. As a bonus, we may consider the space dimension 5. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin–Lions lemma for the compactness argument for its convergence.

SUMMARY: We consider coupled nonlinear Schrödinger (CNLS) equations with a general nonlinearity. We assume that CNLS equations possess a solitary wave of which one component is identically zero and that the pitchfork bifurcation of this solitary wave occurs. Utilizing the Evans function approach, we show that the bifurcated solitary waves are linearly (in fact, orbitally) stable if they are sign-definite and are linearly unstable if they are sign-indefinite. Our assumptions are easier to verify than previous results.

SUMMARY: We consider the derivative nonlinear Schrödinger equation (DNLS) which has \(L^2\)-critical and completely integrable structure. It is known that if the initial data \(u_0\in H^1(\mathbb {R})\) satisfies \(\| u_0\|_{L^2}^2 <4\pi \), the corresponding solution is global and bounded. The main aim of this talk is to characterize this \(4\pi \)-mass condition from potential well theory. We see that the mass threshold value \(4\pi \) gives the turning point in the structure of potential well generated by solitons. Our approach is applicable to more general equation which contains DNLS.

SUMMARY: We consider the Cauchy problems for nonlinear Schrödinger equations with inverse-square potential. \[ i\frac {\partial u}{\partial t} = ( -{\Delta } + V )u + g_{0}(u). \] \(V\in C(\mathbb {R}^{N}\setminus \{0\})\) is assumed the homogeneity of degree \(-2\) and the threshold of the selfadjointness, for example, \(V(x)=-(N-2)^{2}/(4|x|^{2})\). We solve the Cauchy problems in the energy space \(\mathcal {D}=D((1-\Delta +V)^{1/2}) \supsetneq H^{1}(\mathbb {R}^{N})\).

SUMMARY: In this talk we consider the uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We prove that all ground states are positive up to phase rotation, radial, and decreasing. Moreover, by refining the results of Shioji and Watanabe (2016), we prove the uniqueness and nondegeneracy of the positive radial solutions.

SUMMARY: We consider the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV equation. As for the lower nonlinearity model of the KdV equation, Linares, Miyazaki and Ponce prove the local well-posedness under a non-degenerate condition introduced by Cazenave and Naumkin (2017). In this talk, we show that the well-posedness result can be extended into the higher-order equation. We also give a lower bound for the lifespan of the solution. The lifespan depends on two quantities determined by the initial data.

SUMMARY: We consider the KdV type equation which contains the quadratic second derivative nonlinearity. Because the derivative loss occurs from the nonlinear term, the well-posedness in the Sobolev space \(H^s(\mathbb {R})\) cannot be obtained by using the iteration argument. Harrop and Griffiths (2015) proved the well-posedness of this equation in the translation invariant Sobolev space \(l^1H^s(\mathbb {R})\) for \(s>5/2\). To improve this result, we use the gauge transform which was used by Ozawa (1998) for the quadratic derivative nonlinear Schrödinger equation. We prove the well-posedness of the KdV type equation in \(\mathcal {X}^s\) for \(s\ge 1\), where \(\mathcal {X}^s\) is the space of functions in \(H^s(\mathbb {R})\) with bounded primitives.

SUMMARY: We consider the Cauchy problem of the 2D Zakharov–Kuznetsov equation. Our aim is to show the well-posedness in a low regularity Sobolev space. In the proof of the crucial nonlinear estimate resonant interactions appear. Since their shape is very complicated (due to the linear part of Zakharov–Kuznetsov equation), it is challenging to treat all of them. To overcome this, we employ a nonlinear version of the Loomis–Whitney inequality and a suitable Whitney decomposition.

SUMMARY: In this talk, we consider the Cauchy problem for the degenerated Zakharov system. The degeneracy means lack of dispersion in one direction in the Schrödinger equation. In contrast to the Zakharov system, the degenerated Zakharov system is not so much studied yet for complexity of the nonlinear interaction. Barros–Linares (2015) showed local well-posedness of this system in certain Sobolev space by the linear estimate (the Strichartz estimate and the maximal function estimate), so they assume high regularity. The aim of this work is lower the regularity than Barros–Linares in the framework of the Fourier restriction norm method.

SUMMARY: The Cauchy problem for the semilinear Proca equations is considered in the de Sitter spacetime. The effects of the spatial variance are remarked through the properties of the solutions of the problem.

SUMMARY: We consider the Cauchy problem of semilinear diffusion equations in the de Sitter spacetime. We show the asymptotic profiles of the global solutions according to growth order of the nonlinear term and decay property of initial data.

SUMMARY: We consider a two-component system of cubic semilinear wave equations in two space dimensions satisfying the Agemi-type structural condition (Ag) but violating (Ag\(_0\)) and (Ag\(_+\)). For this system, we show that small amplitude solutions are asymptotically free as \(t\to +\infty \).

SUMMARY: We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. Without renormalization of the nonlinearity, we show that solutions to SdNLW with regularized noises tend to \(0\) as the regularization is removed.

SUMMARY: We consider the Cauchy problem for the semilinear wave equation with scale-invariant damping. This equation generalizes the nonlinear wave equation in the FLRW (Friedmann–Lemaitre–Robertson–Walker) spacetime with zero spatial curvature in some case. We show the blow-up phenomena as well as upper bounds of the lifespan of solutions in subcritical and critical cases.

SUMMARY: In this talk, we consider the asymptotic behavior of the global solutions to the initial value problem for the damped wave equation with a nonlinear convection term. We assume that the initial data decay polynomially at spatial infinity. When the initial data decay fast enough, it is known that the solution to this problem converges to a self-similar solution to the Burgers equation called a nonlinear diffusion wave and its optimal asymptotic rate is obtained. In this talk, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.

SUMMARY: We study the asymptotic behavior of the solution to a generalized Rosenau equation that is of regularity-loss type. Due to its structure, the solution behaves differently from the solutions of wave equations with a lower order damping term. In this talk, the author gives a new expanding method for the solution in the high-frequency region.

SUMMARY: We consider a boundary value problem of the stationary transport equation in a two dimensional bounded convex domain with the incoming boundary condition. In this talk, we give a \(W^{1, p}\) estimate of the solution to the boundary value problem with \(1 \leq p < p_m\), where \(W^{1, p}\) is the standard Sobolev space and \(p_m\) is a real number depending only on the shape of the domain. Moreover, we show two examples which implies that this estimate is optimal in some cases. This \(W^{1, p}\) estimate for the solution is important when we discuss reliability of numerical solutions to the boundary value problem obtained by discrete-ordinate discontinuous Galerkin methods.

SUMMARY: Consider the motion of a viscous fluid past a rotating body in 3D, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the linearized non-autonomous system in a fixed exterior domain. We develop \(L^q\)-\(L^r\) decay estimates of the evolution operator generated by this system. Our theorem completely recovers those estimates for the autonomous case (Stokes, Oseen, ...).

SUMMARY: In this talk, we develop the R-boundedness for the generalized Stokes resolvent problem in an infinite layer, with Neumann boundary condition on both upper and lower boundary. This has not been proved for such a boundary condition, while it has been proved for Neumann and Dirichlet boundary condition on upper and lower boundary, respectively. As an application, we also establish the local well-posedness for the incompressible Navier–Stokes equation in an infinite layer with a free surface for both upper and lower boundaries.

SUMMARY: We show the best constant of Hardy–Leray inequality for solenoidal (i.e., divergence-free) fields in \(\mathbb {R}^N\). This is a complement of the former works by O. Costin and V. Maz’ya on sharp Hardy–Leray inequality for axisymmetric divergence-free fields. It turns out from our result that the assumption of axisymmetry can be removed.

SUMMARY: We show that the Stokes operator defined on \(\mathrm {L}^p_{\sigma } (\Omega )\) for an exterior Lipschitz domain \(\Omega \subset \mathbb {R}^n\) \((n \geq 3)\) admits maximal regularity provided that \(p\) satisfies \(\lvert 1/p - 1/2 \rvert < 1/(2n) + \varepsilon \) for some \(\varepsilon > 0\). In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on \(\mathrm {L}^p_\sigma (\Omega )\) for such \(p\). This enables us to prove the existence of mild solutions to the Navier–Stokes equations in the critical space \(\mathrm {L}^{\infty } (0 , T ; \mathrm {L}^3_{\sigma } (\Omega ))\).

SUMMARY: We consider the linearized problem for the Navier–Stokes equation around the solution for the Rayleigh–Plesset equation. Here the Rayleigh–Plesset equation is an ordinary differential equation with respect to time whose solution describe the dynamics of spherical bubble. Since the Rayleigh–Plesset equation is derived from the Navier–Stokes equation, we can describe one of the solution of the Navier–Stokes equation by the solution of the Rayleigh–Plesset equation. Then we show a local existence of the unique solution of the linearized problem for Navier–Stoeks equation around the solution of Rayleigh–Plesset equation.

SUMMARY: We consider the compressible Navier–Stokes–Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier–Stokes system. This enables us to apply Banach’s fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the \(L^2(\mathbb {R}^d)\)-framework.

SUMMARY: We consider the initial boundary value problem for a compressible viscoelastic system with time-periodic external force in an infinite layer. There exists a time-periodic parallel flow if the external force has a suitable condition. We show that if the initial perturbation is sufficiently small, the time-periodic parallel flow is asymptotically stable, provided that the Reynolds and the Mach numbers are small and the propagation speed of the shear wave is large.

SUMMARY: We consider 2-dimensional doubly diffusive convection problem for artificial compressible system. The incompressible Navier–Stokes system is obtained as a singular limit with zero Mach number which is included in the artificial compressible system. It is known for the incompressible system that if the bifurcation parameter increases beyond a certain critical value, then the motionless state becomes unstable and a time periodic flow bifurcates. In this talk, we show that there also exists a bifurcating time periodic solution for the artificial compressible system when the Mach number is sufficiently small.